MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lukshefth1 Structured version   Visualization version   GIF version

Theorem lukshefth1 1767
Description: Lemma for renicax 1769. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
lukshefth1 ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))

Proof of Theorem lukshefth1
StepHypRef Expression
1 lukshef-ax1 1766 . 2 ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏)))))
2 lukshef-ax1 1766 . . . 4 ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜏) ⊼ ((𝜏𝜃) ⊼ (𝜏𝜃)))))
3 lukshef-ax1 1766 . . . 4 (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜏) ⊼ ((𝜏𝜃) ⊼ (𝜏𝜃))))) ⊼ ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))))) ⊼ ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏)))) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))))))))
42, 3nic-mp 1743 . . 3 ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏)))) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))))))
5 lukshef-ax1 1766 . . 3 (((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏)))) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏)))))) ⊼ (((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜓𝜒)) ⊼ (𝜑 ⊼ (𝜓𝜒)))) ⊼ (((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))))) ⊼ (((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒))) ⊼ ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))))))
64, 5nic-mp 1743 . 2 (((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))))) ⊼ (((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒))) ⊼ ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))))
71, 6nic-mp 1743 1 ((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wnan 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 383  df-nan 1595
This theorem is referenced by:  lukshefth2  1768  renicax  1769
  Copyright terms: Public domain W3C validator